So given a velocity from a newtonian problem ex: 5 m/s I can convert to its energy equivalent in special relativity via this formula. Note that as newtonian velocity goes to infintiy relativistic velocity approaches c and at 0 both quantities are 0.

Given this framework I know for fact from earlier that newtonian velocity is given as:

2 Answers
2

In special relativity, proper acceleration is defined as
$$
a = \frac{du}{dt},
$$
where
$$
u = \frac{dx}{d\tau} = v\frac{dt}{d\tau}
$$
is the proper velocity, and
$$
d\tau = dt\sqrt{1-v^2/c^2}
$$
is the proper time. So
$$
\frac{d}{dt}\left(\frac{v}{\sqrt{1-v^2/c^2}}\right) = a.
$$
If we integrate this over a time interval $[0,t]$, we get, if $a$ is constant,
$$
\frac{v}{\sqrt{1-v^2/c^2}} - \frac{v_0}{\sqrt{1-v_0^2/c^2}} = at,
$$
with $v_0$ the initial velocity. If we define the constant
$$
w_0 = \frac{v_0}{\sqrt{1-v_0^2/c^2}},
$$
then
$$
v^2 = (1 - v^2/c^2)(at+w_0)^2,
$$
so that we finally get
$$
v(t) = \frac{at+w_0}{\sqrt{1+(at+w_0)^2/c^2}}.
$$

In the context of Special Relativity, you do need to be careful about constraints such as "assume constant acceleration" without further qualification because, just as there is a need to distinguish between proper time and coordinate time, one must distinguish between proper acceleration (acceleration measured by an accelerometer) and coordinate acceleration, $\ddot x$.

Proper acceleration, like proper time, is frame invariant while coordinate acceleration is not.

It is perfectly acceptable to specify constant proper acceleration but, as you noted, specifying constant coordinate acceleration is inconsistent with Relativistic mechanics.

A simple problem is to solve the motion of a body which accelerates
constantly. What does this mean? We don't mean that its acceleration
as measured by an inertial observer is constant. We mean that it is
moving so that the acceleration measured in an inertial frame
travelling at the same instantaneous velocity as the object is the
same at any moment. If it was a rocket and you were on board you
would experience a constant G force. This problem can be solved in a
number of ways. One is to use four-vector acceleration along its
worldline which must have constant magnitude. Alternatively, the
object is passing constantly from one inertial frame to another in
such a way that its change of speed in a fixed time interval seen as a
Lorentz boost is always the same. From our understanding of adding
velocities we can see that the rapidity r of the object must be
increasing at a constant rate a with respect to the proper time of the
object T. The rapidity is related to velocity v by the equation